Cascaded higher order filter with low sensitivity to component values and a method for designing the same

ABSTRACT

The present invention provides a higher order filter with low sensitivity to component values and to a method of designing the same. The higher order filter is constructed of second and/or third order filter circuits which have been desensitized with respect to component values and gain by impedance tapering the filter circuits. The filter circuits are impedance tapered by successively scaling the resistors and/or capacitors from left to right to minimize loading on the first filter section and to maximize the quality factor of the passive pole. The impedance of the filter sections is increased from left to right by increasing the resistor values and/or by decreasing the capacitor values from left to right. Once the lower order filter circuits have been designed, they are cascaded to provide a higher order filter with low sensitivity to component values and gain.

CROSS REFERENCE TO RELATED APPLICATION

This application claims priority to and the benefit of the filing data of copending provisional application entitled A CASCADED HIGHER ORDER FILTER WITH LOW SENSITIVITY TO COMPONENT VALUES AND A METHOD FOR DESIGNING THE SAME, assigned Ser. No. 60/032,809, filed Dec. 11, 1996 (Attorney Docket Number 60704-8030). The above-referenced provisional application is incorporated herein by reference.

FIELD OF THE INVENTION

The present invention relates to a higher order filter with low sensitivity to component value variations and to a method for designing such filters. More particularly, the present invention relates to a cascaded higher order filter which has been desensitized with respect to component values by cascading impedance-tapered second and/or third order filters to increase the impedance of the filter from left to right, thereby maximizing the quality factor of the passive pole.

BACKGROUND OF THE INVENTION

It is known to minimize the sensitivity of filter characteristics to values of the gain elements of a filter by minimizing the gain-sensitivity product (GSP) of the filter. For example, "Active Filter Design Handbook", by G. S. Moschytz and P. Horn, 1981, discloses using a computer program to design biquad active filters with a low GSP. The computer program presents flow-chart listings to the designer and allows the designer to enter the specifications for the filter in terms of pole and zero frequencies and quality factors. In response to the data input by the designer, the computer program causes the minimum gain-sensitivity product (GSP) to be output. However, since the resistor values must be selected so as to minimize the GSP, this method results in only partial impedance tapering, namely, impedance tapering with respect to the capacitors of the circuit. Therefore, this method only minimizes the sensitivity of the filter characteristics to values of the gain elements.

Accordingly, a need exists for a method of designing higher order cascaded filters which minimizes the sensitivity of the filter characteristics with respect to component values and gain, thus providing a greater degree of freedom to the filter designer in selecting the components used to construct the filter.

SUMMARY OF THE INVENTION

The present invention provides a method for designing higher order (greater than third order) cascaded filters with low sensitivity to components, including gain elements, by cascading second and/or third order impedance-scaled filters to minimize loading on previous sections of the filter and to maximize the pole quality factor of the passive section of the filter. In accordance with the present invention, the sensitivity of a second order filter to component values and gain is minimized by scaling the impedance of the second filter section of the filter circuit upwards with respect to of the first section so as to minimize the loading on the first filter section. By impedance scaling the filter circuit in this manner, the pole quality factor of the first filter section is maximized. For third-order filters, the sensitivity of the filter with respect to the component values and gain is minimized by impedance scaling the filter sections from left to right in a manner similar to that for the second order filter to minimize loading on the first filter section and to increase the pole quality factor thereof. In accordance with the preferred embodiment of the present invention, filters of higher order than three having low sensitivity to component values and gain are constructed by cascading impedance-tapered second and/or third-order filters which have been impedance tapered in accordance with the method of the present invention.

The method of the present invention is particularly advantageous in designing higher order filters because impedance tapering reduces the number of amplifiers needed in designing the filters which in turn reduces the overall power consumption of the filter.

Accordingly, it is an object of the present invention to provide a cascaded higher order filter with minimum sensitivity to component values.

It is another object of the present invention to provide a filter which can be designed and manufactured in a relatively short period of time due to greater freedom in selecting the circuit components.

It is another object of the present invention to provide a method for designing discrete component active RC filters which have low sensitivity to component value variations.

It is yet another object of the present invention to provide a filter with relatively wide-tolerance components which does not require tuning.

It is yet another object of the present invention to provide a filter with low sensitivity to component values which can be comprised of discrete components or, alternatively, which can be manufactured as an integrated circuit.

It is yet another object of the present invention to provide a filter which can be constructed of discrete components and which is particularly well suited for use with digital signal processors.

It is yet another object of the present invention to provide a filter which can be constructed at relatively low cost.

It is yet another object of the present invention to provide a filter which has low power consumption.

These and other objects of the present invention will become apparent from the following description, drawings and claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A and 1B illustrate the root loci in the s-plane for sixth-order Butterworth and Chebychev filters, respectively.

FIGS. 2A and 2B illustrate the amplitude responses for the sixth-order Butterworth and Chebychev filters having the root loci shown in FIGS. 1A and 1B, respectively.

FIG. 3 illustrates the loss specifications for a third-order lowpass active RC filter.

FIGS. 4A and 4B illustrate the frequency responses for a third-order Butterworth filter and a third-order Chebychev filter, respectively.

FIGS. 5A and 5B illustrate the frequency responses for a sixth-order Butterworth filter and a sixth-order Chebychev filter, respectively.

FIG. 6 graphically compares the ripple for Butterworth and Chebychev filters as a function of second through sixth order as a function of pole quality factor.

FIG. 7 illustrates an nth-order allpole single-amplifier filter circuit.

FIG. 8A illustrates a second order lowpass filter circuit comprising an ideal noninverting voltage amplifier.

FIG. 8B illustrates a second order lowpass filter circuit comprising an ideal noninverting operational amplifier.

FIG. 9 corresponds to the filter circuit of FIG. 8A redrawn with impedance scaling factors in accordance with the impedance tapering method of the present invention.

FIG. 10 graphically illustrates the effect the impedance tapering method of the present invention has on the pole quality factor.

FIG. 11 illustrates amplitude response curves which demonstrate the effect the impedance tapering method of the present invention has on the sensitivity of filters.

FIG. 12 illustrates a third order lowpass filter circuit.

FIG. 13 illustrates the third order lowpass filter circuit of FIG. 12 impedance tapered in accordance with the method of the present invention.

FIG. 14 illustrates amplitude response curves for the third order filter of FIG. 13 which demonstrate the effect the impedance tapering method of the present invention has on a Butterworth filter and a Chebychev filter.

FIG. 15 illustrates the loss specifications for a lowpass Chebychev filter.

FIG. 16 illustrates a third order lowpass Chebychev filter which has been impedance tapered in accordance with the method of the present invention.

FIG. 17 illustrates the loss specifications for a third order lowpass Butterworth filter.

FIG. 18 illustrates the loss specifications for a third order lowpass Chebychev filter.

FIG. 19 illustrates frequency response curves for the Butterworth and Chebychev filters having the loss specifications shown in FIGS. 17 and 18, respectively, which demonstrate the effect of impedance tapering only the resistors of the circuit.

FIG. 20 illustrates a third order filter which has been impedance tapered in accordance with the method of the present invention.

FIG. 21 illustrates functions of the tapering factors verses design frequency.

FIG. 22 illustrates amplitude response curves for impedance tapered third order filters which demonstrate the effect the design frequency has on the filter sensitivity.

FIG. 23 illustrates the graphical method for finding ω₀ for the case of r₂ ≈r₃.

FIGS. 24 and 25 are tables of equations which express the coefficient sensitivities of for al and a₂, respectively, to component variations for third order lowpass filters.

FIG. 26 is a table of equations which express the relationship between a_(i) and ω_(pi), q_(pi) and γ for allpole functions of order n=2 to 6.

FIG. 27 is a table of design values and parameters for the third order filter having the amplitude responses shown in FIG. 14.

FIG. 28 is a table listing the design values and parameters for resistance-tapered third order filters having the loss specifications shown in FIGS. 17 and 18.

FIG. 29 is a table containing values which indicate the dependence of the design parameters r₂, r₃, r₃ /r₂ and β on the selection of ω₀.

FIG. 30 illustrates a fourth order filter having a configuration similar to the third order filter of FIG. 13 and the nth order filter of FIG. 7.

DETAILED DESCRIPTION OF THE INVENTION

The present invention is particularly well suited for, but not limited to, designing discrete-component, cascaded, active RC filters due to the freedom the method of the present invention provides in selecting filter components. Discrete component active RC filters have a distinct edge over their high-tech counterparts (e.g., Digital and DSP, switched-capacitor and switched-current, continuous-time IC) in that the former provides, for example: (1) fast turn-around time for design and manufacture; (2) low power requirements (e.g., only one operational amplifier required per filter); (3) low cost (e.g., no need for AD/DA converters, anti-aliasing filters, etc.); and (4) moderate size (i.e., smaller than LC filters but larger than IC chips).

In accordance with the present invention, a method for designing higher order cascaded filters is provided which allows discrete-component active RC filters to be manufactured with relatively wide-tolerance RC components while maintaining the cost-effectiveness of these types of filters. However, it will become apparent to those skilled in the art after reading the present application that the impedance tapering method of the present invention is also applicable and is equally advantageous to filters manufactured as integrated circuits. It will also become apparent to those skilled in the art that, although the present invention is described below with respect to active-RC allpole filters, the present invention is not limited to allpole filters, and that the present invention is applicable to filters having both poles and zeros as well as to passive filters. Furthermore, it will become apparent to those skilled in the art that the present invention is not limited to any particular types of filters. The method of the present invention can be used to design, for example, lowpass filters, bandpass filters, band-reject filters, highpass filters, notch filters and state-variable biquad filters. The method of the present invention can also be used to design specific types of these filters, such as, for example, Butterworth filters, Chebyshev filters, Bessel filters, Optimal filters, Parabolic filters and Elliptic filters.

In the interest of brevity, however, the present invention will be described only with respect to Butterworth and Chebyshev filters and, more particularly, with respect to active RC allpole lowpass Butterworth and Chebyshev filters. However, it will become apparent to those skilled in the art after reading the present application that the present invention is in no way limited to the filters discussed below. It will also become apparent to those skilled in the art after reading the present application that the present invention is not limited to any particular order filter and that the method of the present invention can be used to design low sensitivity filters of any order.

The present invention will now be described with respect to the preferred embodiments. In designing low-sensitivity active RC allpole filters, the filter designer generally has three factors to consider:

(i) Component sensitivity increases with inband ripple. Thus, the specifications should be geared towards a low-ripple (or, if possible, maximally-flat) amplitude response. This, in turn, decreases the pole quality factors (Qs);

(ii) Component sensitivity increases with the filter order. The latter should therefore be held as low as possible consistent with the filter specifications. Fortunately, this is standard procedure in filter design and minimizes filter cost;

(iii) Impedance tapering minimizes sensitivity. Using the impedance tapering method of the present invention, the filter circuit can be directly and significantly desensitized with respect to component values and gain at no extra cost. All that is required is that the designer make a judicious choice of component values.

Since it is well known that factors (i) and (ii) must be taken into consideration, and since the present invention is primarily directed to the factor set forth in paragraph (iii), only factor (iii) will be discussed in detail below.

In accordance with the present invention, it has been determined that the larger the impedance tapering factors can be made, the less sensitive the filter circuit will be to component values, i.e., the circuit will have a higher tolerance to variations of the component values. However, the impedance tapering factor cannot be made arbitrarily large because the maximum possible degree of impedance tapering depends directly on the values of the transfer function coefficients. Therefore, in accordance with the present invention, bounds on the impedance tapering factor as a function of the transfer function coefficients will be provided below, both for second and third-order lowpass filters. Sensitivity expressions and design equations are also given below for these filters and data from P SPICE Monte Carlo simulations is provided which shows the effectiveness of the impedance tapering method of the present invention as a means of reducing filter sensitivity to component values and amplifier gain. Because the impedance tapering factors cannot be chosen arbitrarily (since they depend on the transfer function coefficients), a detailed design procedure for selecting the impedance tapering factors is provided below.

Before describing the design procedure of the present invention, sensitivity expressions which define the sensitivity of the transfer function coefficients to component values will be derived and the sensitivity of filters to component values will be shown. In order to demonstrate the sensitivity of filters to component values and to filter order, characteristics of second order, third order, sixth order and nth order filters will be investigated. Once the sensitivity of filters to component values and to filter order has been demonstrated, the sensitivity expressions will be used to derive design equations for designing second and third order filters with low sensitivity to component values and gain. The method of the present invention for designing second and third order filters will then be described in detail. In accordance with the preferred embodiment of the present invention, higher order filters with low sensitivity to component values and gain are designed by cascading second and/or third order filters which have been impedance tapered in accordance with the impedance tapering method of the present invention.

SENSITIVITY OF THE TRANSFER FUNCTION TO COEFFICIENT VARIATIONS

The transfer function T(s) of an nth-order, allpole lowpass filter is: ##EQU1## The frequency response of the filter depends on the coefficients a_(i) of the polynomial D(s). These coefficients are available from any filter handbook or CAD program and determine the location of the poles (i.e., the roots of D(s)) in the s-plane. β corresponds to the gain of the amplifier of the filter. For example, the poles of a sixth-order Butterworth lowpass filter will lie on a semicircle about the origin in the left half plane and those of a Chebyshev filter will lie on an ellipse. The root loci for these particular filters are shown in FIGS. 1a and 1b, respectively. The larger the ripple of the Chebyshev filter, the smaller the eccentricity ε will be (i.e., the closer the poles will be to the jω axis) and the higher the corresponding pole Qs will be. It should be noted, however, that the slope of the asymptotic responses of these two allpole filters of equal order n will be the same, irrespective of the inband ripple. This is shown in the corresponding amplitude responses shown qualitatively in FIGS. 2A and B, respectively. The ripple of the Chebyshev filter is equal to (1+e²)^(-1/2) where the ellipse eccentricity ε=tanh (1/n sinh⁻¹ 1/e).

To obtain the sensitivity of the transfer function T(s) (equation 1) to variations of the circuit components, some basic relations in sensitivity theory will be briefly discussed. The relative sensitivity of a function F(x) to variations of a variable x is defined as ##EQU2## Thus, the relative variation of F(x) caused by a relative change of x is given by ##EQU3## where the difference values ΔF and Δx, rather than their differentials dF and dx, are used to imply measurable relative quantities, e.g., as given in percentages of their nominal values. The amplitude and phase sensitivity expressions for T(s) are obtained by letting s=jω, where

    T(s)|.sub.s=jω =|T(jω)|·e.sup.jφ(ω)

    ln T(jω)=ln |T(jω)|+jφ(ω)(Eq. 4)

and

    =α(ω)+jφ(ω)                          (Eq. 5)

α(ω) is the amplitude response in Nepers, and α(ω) is the phase response in degrees. Thus, with equation 2, the following is obtained: ##EQU4## With the symbol ##EQU5## signifying the semi-relative sensitivity of a function F(x) to a relative change in x. The amplitude sensitivity and phase sensitivity expressions for T(s) are given by Equations 8 and 9, respectively, as follows:

    Re{S.sub.x.sup.T(jω) }=S.sub.x.sup.α(ω)  (Eq. 8)

    Im{S.sub.x.sup.T(jω) }=S.sub.x.sup.φ(ω)    (Eq. 9)

The amplitude sensitivity in Equation 8 is given in Nepers N! per relative change in a real variable x, i.e., ##EQU6## Converting the amplitude sensitivity from Nepers to dB per percent component change, the following expression is obtained: ##EQU7## Thus, for example, an amplitude variation of 0.1 N corresponds to 0.868 dB, or an amplitude variation of approximately 1 dB.

From the sensitivity expressions presented above, it follows that the relative change of a transfer function T(s), as given, for example, in equation 1, to the variations of its coefficients a_(i), can be written as: ##EQU8##

The relative sensitivity function defined by equation 2 is very useful because with it, the sensitivity of many commonly occurring function types can readily be obtained and compared. To do so, the following simple and easily derivable relationships are used:

    S.sub.x.sup.u(x)·v(x) =S.sub.x.sup.u(x) +S.sub.x.sup.v(x)(Eq. 13)

    S.sub.x.sup.u(x)/v(x) =S.sub.x.sup.u(x) -S.sub.x.sup.v(x)  (Eq. 14) ##EQU9##

    S.sub.x.sup.c·y(x) =S.sub.x.sup.y(x)              (Eq. 16)

    S.sub.x.sup.c·x =1                                (Eq. 17)

    S.sub.x.sup.y.spsp.n.sup.(x) nS.sub.x.sup.y(x)             (Eq. 18)

    S.sub.x.spsb.n.sup.y(x) =1/n=S.sub.x.sup.y(x)              (Eq. 19)

With equations 14, 15 and 16, it readily follows from equation 1 that

    S.sub.a.sbsb.i.sup.T(s) =S.sub.a.sbsb.i.sup.N(s) -S.sub.a.sbsb.i.sup.D(s) i=0, 1, 2, . . . n                                        (Eq. 20)

Thus ##EQU10## and for i≠0 ##EQU11## Letting ##EQU12## and, with equation 21 ##EQU13## the following is obtained: ##EQU14##

At this point, two important properties of the coefficients a_(i) of the denominator polynomial D(s) of a filter are recalled, namely:

(i) the coefficients a_(i) of D(s) must be real (this applies also to the numerator polynomial) resulting in either real or complex-conjugate poles (and zeros); and

(ii) the coefficients a_(i) of D(s) must be positive (this is one of the conditions for a Hurwitz polynomial) resulting in only left-half plane (LHP) poles (i.e. D(s) must be a Hurwitz polynomial).

From point (i) above, it follows that the Δa_(i) /a_(i) terms in equation 25 are real, so that this expression can be rewritten as: ##EQU15## where, for i=1, 2, . . . n ##EQU16##

The functions f_(i) (ω) are frequency-dependent multiplicands of the coefficient variations Δa_(i) /a_(i) which cause the amplitude deviation Δα(ω). They depend only on the initial transfer function T(s) of a given filter, or, more specifically, on its denominator coefficients a_(i).

The third-order allpole transfer function given in terms of the polynomial coefficients a_(i) (i=0, 1, 2) is: ##EQU17##

With the negative-real pole γ and the complex-conjugate pole pair given in terms of the pole frequency ω_(p) and the pole Q, q_(p), this becomes: ##EQU18##

This describes a third-order lowpass filter where β is the dc gain of its active-RC realization. Assuming that the required filter loss specifications are those given in FIG. 3, the frequency bands correspond to those of a transmit lowpass filter used in ADSL (Asymmnetrical Digital Subscriber-Loop) data transmission systems. Typically A_(max) 0.5 dB, A_(min) =25 dB, and β=4. For these specifications, the filter coefficients given in Table 1 and die γ, ω_(p) and q_(p) values given in Table 2 are obtained (e.g., from filter tables or a CAD filter program) as follows.

                  TABLE 1     ______________________________________     Coefficients of 3rd-order lowpass filter     Filter Type     Passband ripple (dB)                  a.sub.2   a.sub.1    a.sub.0     ______________________________________     Butterworth (0.5)                  1.43 × 10.sup.6                            1.02 × 10.sup.12                                       3.64 × 10.sup.17     Chebyshev (0.5)                   6.3 × 10.sup.6                            3.88 × 10.sup.11                                       9.09 × 10.sup.16     ______________________________________

                  TABLE 2     ______________________________________     Pole frequency and Q (i.e. ω.sub.p and q.sub.p) for the poles of a     3rd order     lowpass filter with denominator     Filter Type                γ      ω.sub.p                                         q.sub.p     ______________________________________     Butterworth                713.727 × 10.sup.3                             713.727 × 10.sup.3                                        1.0     (0.5 dB ripple)     Chebyshev  314.891 × 10.sup.3                             537.264 × 10.sup.3                                        1.71     (0.5 dB ripple)     ______________________________________

Calculating the functions (see eqs. (27) and (28)): ##EQU19## and where a₃ =1, the functions plotted in FIGS. 4A and 4B are obtained. Note that the highest sensitivity of the amplitude response α(ω) to coefficient variations is just beyond the cutoff frequency of 80 KHz. Moreover, the sensitivity of the Chebyshev transfer function to coefficient variations is larger than that of the Butterworth filter (the maximum occurs for f₀ (ω)) Butterworth! with approximately 1.6, and for f₁ (ω) Chebyshev! with approximately 2.0). The distribution of the three functions along the frequency axis for the two filters is similar, primarily because the pole locations of the two filters do not differ greatly. Within the passband, i.e., below 80 KHz, the sensitivity curve of the a₂ coefficient, i.e., f₂ (ω)), is lowest for both filters whereas for the a₁ and a₀ coefficients (i.e., f₁ (ω) and f₀ (ω) it is approximately the same.

The transfer function of a 6th-order Butterworth and Chebyshev allpole lowpass filter is expressed as: ##EQU20##

Assuming the filter specifications in FIG. 3 to be A_(max) =0.1 dB and A_(min) =50 dB, the following filter coefficients are obtained:

                  TABLE 3     ______________________________________     Coefficients of 6th-order lowpass filter     Filtertype     Passband     ripple (dB)              a.sub.5 a.sub.4 a.sub.3                                    a.sub.2                                          a.sub.1                                                a.sub.0     ______________________________________     Butterworth              2.66 ×                      3.53 ×                              2.97 ×                                    1.67 ×                                          5.94 ×                                                1.06 ×     (0.1)    10.sup.6                      10.sup.12                              10.sup.18                                    10.sup.24                                          10.sup.29                                                10.sup.35     Chebyshev              8.61 ×                      7.49 ×                              3.53 ×                                    1.31 ×                                          2.89 ×                                                3.34 ×     (0.1)    10.sup.5                      10.sup.11                              10.sup.17                                    10.sup.23                                          10.sup.28                                                10.sup.33     ______________________________________

Calculating the corresponding coefficient-sensitivity functions f_(i) (ω), for i=0, 1, . . . 5, as in equations 27 and 28, the frequency response plots shown in FIGS. 5A and 5B are obtained. As in the third-order case (FIGS. 4A and 4B), the coefficient sensitivities for the Chebyshev filters are significantly higher (up to double) than those of the Butterworth filters. One reason for this is that the amplitude sensitivity is proportional to the pole quality factors, and these are higher for the Chebyshev than for the Butterworth filter, as a comparison of Tables 2 (above) and 4 (below) shows. In fact, it will be shown below that the sensitivity of the coefficients to component values is also critically dependent on the pole quality factors. It should be noted that for the Butterworth filter, ω_(p) is constant (i.e., 687.598×10³) for the three biquad sections, because the poles of a Butterworth filter lie on a semicircle with radius ω_(p). For the Chebyshev filter, the poles lie on an ellipse. Therefore, the ω_(p) values differ for each biquad section.

                  TABLE 4     ______________________________________     Pole frequency and Q (i.e., ω.sub.p and q.sub.p) for the three pole     pairs of a 6th     order lowpass filter                Section     Filter Type                no.         q.sub.p                                   ω.sub.p     ______________________________________     Butterworth                1           0.518  687.598 × 10.sup.3     (0.1 dB ripple)                2           0.707  687.598 × 10.sup.3                3           1.93   687.598 × 10.sup.3     Chebyshev  1           0.599  257.956 × 10.sup.3     (0.1 dB ripple)                2           1.33   419.460 × 10.sup.3                3           4.63   534.184 × 10.sup.3     ______________________________________

It should be noted that the frequency dependent sensitivity functions f_(i) (ω) plotted in FIGS. 4A-B and 5A-B depend only on the generic filter type (e.g., Chebyshev or Butterworth) but not on the actual circuit used to realize the filter. Thus, they depend on the given filter specifications and on the required filter order. This, in turn, will affect the filter cost, since the number of required filter components generally depends on the filter order. Economics aside, however, the sensitivity functions shown demonstrate a direct dependence of sensitivity on the quality factors of the transfer function poles: the higher the pole quality factors the higher the sensitivity. Therefore, it can be concluded that in order to design a filter with low sensitivity to its component values, the filter with the lowest possible pole quality factors (consistent with the filter specifications) should be used. Thus, for example, with respect to sensitivity, a Butterworth filter is preferable to a Chebyshev filter and, likewise, a low-ripple Chebyshev filter is preferable to a Chebyshev filter with higher ripple. Unfortunately, this preference frequently conflicts with cost, since the lower the ripple, for a given filter specification, the higher the required order will generally be. A Butterworth filter, with its "maximally flat" amplitude response corresponds to the limit case of no ripple in the filter passband and, compared to a Chebyshev filter of equal order, invariably has lower pole quality factors (Qs). This is illustrated in FIG. 6 where the highest pole Q of 2nd to 6th-order Butterworth and Chebyshev filters of varying-ripple is shown. FIG. 6 clearly indicates that in order to keep the pole quality factors at a minimum for the sake of low filter sensitivity to coefficient variations, it is desirable to design the filter with as low ripple and as low order as consistent with the filter specifications.

SENSITIVITY OF THE COEFFICIENTS TO COMPONENT VARIATIONS

As demonstrated above, the variation of the amplitude response α(ω) of an allpole filter (i.e., no finite zeros) caused by changes in the coefficients a_(i) of the corresponding filter transfer function is given by: ##EQU21## where the frequency functions f_(i) (ω) depend only on the type of filter used (e.g., Chebyshev, Butterworth, etc.). On the other hand, the coefficient variations Δa_(i) /a_(i) depend on the circuit type used to design the filter (e.g., positive or negative feedback, etc.) and on the values of the circuit components used to obtain a given transfer function T(s), such as that given by equation 1. Furthermore, it has been shown above that variations in the frequency dependent functions f_(i) (ω) and in the coefficients Δa_(i) /a_(i) will be all the larger the higher the pole Qs of the filter. Consequently, the passband ripple will also be larger the higher the pole Q's are (see FIG. 6).

The sensitivity of the coefficients a_(i) to variations of the circuit components (i.e., the resistors, capacitors, and amplifiers) making up the filter will now be examined. For the purpose of illustration, the representative nth-order general allpole single-amplifier filter structure shown in FIG. 7 will be used to illustrate the sensitivity of the transfer function coefficients to component value variations. This is a lowpass filter, but any other filter (e.g., highpass or bandpass) could also be used for this purpose. The transfer function of this filter will have the form of equation 1. The amplitude variation due to coefficient variations is given by equation 34 above. The coefficients a_(i) are functions of the resistors, the capacitors and the gain β. Thus, using equation 3, the coefficient variations can be expressed in the form: ##EQU22##

In general, the individual resistors R.sub.μ, capacitors C_(v), and gain-determining resistors will be characterized by their mean μ_(x), and standard deviation σ_(x), where x represents each of the component types (e.g., resistors or capacitors). The coefficient variations Δa_(i) /a_(i) will then be random variables whose statistical behavior is a function of the components on which they depend.

In what follows, the coefficient-to-component sensitivity is derived analytically for the second, third and nth-order allpole filters. The resulting expressions provide insight into the methods of the present invention for deterministically reducing this sensitivity. To examine the efficacy of these methods, data obtained from testing using P SPICE Monte Carlo analysis with given component statistics will be provided. It will thus be shown that the method of the present invention for minimizing the sensitivity of filters is indeed very effective in reducing frequency response variations caused by component value variations.

SENSITIVITY EXPRESSIONS FOR THE SECOND ORDER FILTER

FIG. 8A illustrates a second-order lowpass filter. In FIG. 8B, the ideal noninverting voltage amplifier with gain β of FIG. 8A has been replaced by an ideal noninverting operational amplifier. The voltage transfer function for this circuit (known as class 4 or Sallen and Key) expressed in terms of the coefficients a_(i) is given by: ##EQU23## and in terms of the pole frequency ω_(p) and pole Q, q_(p), by: ##EQU24## Considering the overall variation of coefficient a₀, the following is obtained from the general expression of equation 35: ##EQU25## With the sensitivity expressions of equations 13-19 it follows that:

    S.sub.x.sup.a.sbsp.0 =2S.sub.x.sup.ω.sbsp.p          (Eq. 41)

and

    S.sub.R.sbsb.i.sup.a.sbsp.0 =S.sub.C.sbsb..sup.a.sbsp.p =-1 i=1, 2(Eq. 42)

Thus: ##EQU26##

Furthermore, the mean of Δa₀ /a₀ will equal the negative sum of the means of all R.sub.μ and C_(v) and the variance will be the positive sum of their variances. Note that since the sensitivity of a₀ to all components x is -1 (and to the gain β, it is zero), there is nothing that can be done with regard to circuit design that will effect the component sensitivity of a₀. The only possible measures which can be used to decrease Δa₀ /a₀ are technological, i.e., Δa₀ /a₀ can be decreased only by prescribing the quality, precision, temperature coefficient, aging behavior, etc., of the resistors and capacitors. This conclusion is true for all filters of the type shown in FIG. 7, regardless of their order n.

Next, the variation of a₁ due to component values will be considered. From equation 35, the following is obtained: ##EQU27## With Equation 39 and the expressions of equations 13-19, the following is obtained: ##EQU28## In the second part of equation 46, the fact that a₁ =ω_(p) /q_(p) and that ω_(p) =(R₁ R₂ C₁ C₂)^(-1/2) has been utilized. Similarly, for the sensitivities of al to the remaining components, the following is obtained: ##EQU29## The last expression results from equation 38, since ω_(p) is independent of β. Calculating q_(p) from equations 38 and 39, the following is obtained: ##EQU30## where ##EQU31##

    κ.sub.2 =R.sub.1 C.sub.1 +R.sub.2 C.sub.2 +R.sub.1 C.sub.2(Eq. 53)

    κ.sub.3 =R.sub.1 C.sub.1                             (Eq. 54)

Denoting the pole Q of the passive network (i.e. β=0) by q, i.e.,

    q=q.sub.p (β=0)=κ.sub.1 /κ.sub.2          (Eq. 55)

equation 56 is obtained from the sensitivity expressions of equations 13-19: ##EQU32##

The coefficient sensitivities given above are all proportional to the pole quality factor, q_(p). Thus, in deciding on a filter type for a given application, preferably the one yielding the lowest pole Qs will be selected. This means that a low ripple or maximally flat filter should be used if possible, with as low a filter degree n as the specifications will allow (see FIG. 6).

From equations 56 and 57 it follows that the coefficient (or q_(p)) sensitivity to the gain is inversely proportional to the passive RC pole quality factor, q. Thus, once the filter type has been chosen, it should be designed so that the passive pole quality factor, q, is as large as possible. Since a passive RC network can have only negative-real, single poles, it follows that q is limited to less than 0.5. The upper bound of 0.5 cannot actually be reached, since this entails a component spread approaching infinity. This can be easily shown for the circuit of FIG. 7. From equations 51-54, the following expression is obtained: ##EQU33## Inverting this expression, the following is obtained: ##EQU34## Letting ##EQU35## equation 59 takes on the form: ##EQU36##

Since the function y=x+x⁻¹ reaches its minimum value, namely 2, for x=1, it follows from equation 61 that for R₁ C₁ =R₂ C₂ : ##EQU37##

This quantity approaches its minimum value only when the capacitor spread C₂ /C₁ approaches zero. In other words, the maximum value q_(max) =0.5 is reached only in the limit, as C₂ /C₁ approaches zero and R₁ C₁ =R₂ C₂.

This result can be generalized as follows. It can be shown that the pole Q, q of an RC ladder-type structure can be maximized by impedance scaling the second L-section of the ladder so as to minimize the loading on the first L section. For example, referring to FIG. 8A, the second L-section comprising R₂ and C₂ can be impedance scaled upwards so as to minimize the loading on the first, i.e., on R₁ and C₁. Referring to FIG. 9, in which the active RC lowpass filter of FIG. 8A has been redrawn with impedance scaling factors r and ρ for R₂ and C₂, respectively, i.e.,

    R.sub.1 =R; C.sub.1 =C

    R.sub.2 =r·R; C.sub.2 =C/ρ                    (Eq. 63)

the following is obtained: ##EQU38## and for r=ρ: ##EQU39##

Thus, impedance scaling R₂ and C₂ by a value of p>1, q will approach 0.5 and the sensitivity of a₁ (or q_(p)) to β will, according to equation 56, be minimized. In accordance with the present invention, it has been determined that ρ does not have to be that much larger than unity to be effective, as the plot of q versus ρ shows in FIG. 10. This is fortunate since, in practice, a large ρ may cause C₂ to decrease into the range of the parasitic capacitance of the circuit. From FIG. 10 it is apparent that a value of ρ between 2 and 3 will bring q close to its upper boundary of 0.5.

IMPEDANCE TAPERING THE SECOND ORDER FILTER

It will now be shown that impedance scaling with a factor ρ>1 will also decrease the other coefficient sensitivities, given by equations 46-50. Inserting the expressions in equation 63 into equations 46 through 50, the sensitivity relations given below are obtained. Although ideal impedance scaling requires that r=ρ, this may not always be possible, since a given design may require two degrees of freedom, i.e., freedom in choosing both ρ and r. Both sets of expressions, namely those for r=ρ and for r≠ρ, are given below. ##EQU40##

Clearly, ideal impedance scaling (in which case ρ=r>1) also reduces the coefficient sensitivities to the gain β but also to the other components. If r is required to be unequal to ρ (for reasons of design flexibility), then increasing only the capacitor ratio ρ or, alternatively, increasing only r, will also reduce the sensitivities. Even though equations 67 and 69 include a term (ρ/r)^(1/2) (1-β), this term will be small or even zero, since the gain β will generally be in the range between unity and two.

To demonstrate the effect of the impedance tapering method of the present invention, FIG. 11 shows P SPICE Monte Carlo runs of the circuit shown in FIG. 9 for ρ values ranging from 0.1 to 10, and q_(p) values from 1 to 5. Comparing scenarios a), b) and c) in FIG. 11 for ρ=r=1, the influence of the pole Q on component sensitivity for non-impedance scaled circuits is shown. As the pole quality factor is increased, the circuit becomes increasingly sensitive to component variations, which are uniformly distributed with zero mean and 5% tolerance. By impedance scaling by a factor of only 3 (i.e. ρ=r=3), the sensitivities are decreased significantly. For p=r=10, the sensitivities are decreased even more. Conversely, when ρ=r=0.1, the high sensitivity of the circuits renders them practically useless.

SENSITIVITY EXPRESSIONS FOR THE THIRD ORDER FILTER

The third-order version of the nth-order lowpass filter shown in FIG. 7 is shown in FIG. 12. The voltage transfer function is given by: ##EQU41## or, in terms of the pole frequencies ω_(p), and γ the pole Q, q_(p) : ##EQU42##

Note that γ is equal to ω_(p) for a Butterworth, and to the eccentricity ε times ω_(p) for a Chebyshev third-order lowpass filter. The coefficients of T(s) in terms of the circuit components are given by: ##EQU43## where, in terms of the eccentricity ε, ##EQU44## With equation 34 and the sensitivity expressions given by equations 13-19, the following is obtained: ##EQU45## For the case of a Butterworth filter, where γ=ω_(p), it follows that ##EQU46## The variation of a₁ and a₂ follow from equation 34 as: ##EQU47## Similarly, the sensitivity of a₂ to all the circuit components follows as: ##EQU48##

The purpose of deriving the sensitivities above is to examine the fact that they can lead to design guidelines which ensure that the individual coefficient sensitivities will be minimized. It was discussed above with respect to second-order ladder filters that impedance scaling the second L section upwards to decrease the loading on the first L section increases the passive RC pole quality factor, q, towards its absolute, but non-attainable, maximum of 0.5. This decreases not only the q_(p) sensitivities with respect to variations of the gain β, but also the coefficient sensitivities with respect to the RC components. For filters of degree higher than two it is no longer practical to derive expressions for individual poles as functions of the components. Thus, in the case of n>3, the closed-form expressions for γ, ω_(p), and q_(p) as a function of the resistors, capacitors and gain β of the circuit, cannot be obtained, nor can an expression for a quality factor related to the passive RC ladder network be obtained. Nevertheless, analytical expressions for the coefficient sensitivity to individual component values, as well as numerical computations, indicate that impedance scaling the ladder network of an nth-order filter, such as that shown in FIG. 7, with a scaling factor ρ>1 reduces the overall sensitivity of the network to all component values.

For the third-order network, this results in the circuit shown in FIG. 13, where each resistor R_(i) is increased from left to right by a factor n, and each capacitor C_(i) is decreased by a factor ρ_(i) ⁻¹. Preferably, r₁ is equal to ρ_(i) for best impedance scaling. However, in order to maintain a sufficient number of degrees of freedom, this condition cannot, in general, be satisfied in the case of the third-order filter. Nevertheless, because the impedance level is being gradually increased, L-section by L-section, from left to right, this process of the present invention shall be referred to herein as network impedance tapering the ladder network, or simply tapering. By tapering in accordance with the present invention, the impedance of the resistors, capacitors, or both, can be increased from left to right.

IMPEDANCE TAPERING THE THIRD ORDER FILTER

Referring to FIG. 13 ideal tapering requires that:

    r.sub.2 =ρ.sub.2 =ρ

    r.sub.3 ρ.sub.3 =ρ.sup.2                           (Eq. 93)

And for the general nth-order network shown in FIG. 7, ideal tapering requires that

    r.sub.1 =ρ.sub.i =ρ.sup.i-1                        (Eq. 94)

It will now be shown that the coefficient sensitivities of the third-order network shown in FIG. 13 can be reduced by impedance tapering. By inserting the general impedance scaling factors r₁ and ρ_(i) (as shown in FIG. 13) into the sensitivity expressions given by equations 79-92:

    R.sub.1 =R; R.sub.2 =r.sub.2 R; R.sub.3 =r.sub.3 R

    C.sub.1 =C; C.sub.2 =C/ρ.sub.2 ; C.sub.3 =C/ρ.sub.3(Eq. 95)

the expressions listed in the first column of the tables shown in FIGS. 24 and 25, are obtained. Tapering the resistors separately by

    n=r.sup.i-1                                                (Eq. 96)

and the capacitors by

    ρ.sub.i =ρ.sup.i-1                                 (Eq. 97)

the sensitivity expressions in column 2 of the two tables of FIGS. 24 and 25 are obtained. Finally, for ideal impedance tapering, the condition of equation 94, which results in the third column of the tables in FIGS. 24 and 25, is obtained.

It can be shown that in general, the coefficient sensitivity of the impedance tapered filter has the form: ##EQU49## where x is any resistor, capacitor or gain element of the circuit. For ideal impedance tapering, (see eq. 94) it follows that:

    a.sub.0 (RC).sup.i =1                                      (Eq. 99)

and ##EQU50##

Whereas the coefficients a_(i) are given by the filter specifications, the function f(ρ) can readily be minimized by making ρ>1. Here again, as with the second order filter, a ρ between 2 and 3 will reduce the coefficient sensitivity appreciably. The same applies to the nonideally tapered circuits for which the coefficient sensitivities are given in the first two columns of the tables shown in FIGS. 24 and 25.

In practice, the filter specifications often do not permit ideal impedance tapering, in which case just the capacitors may be tapered with a ρ value of, for example, between 2 and 3, and the resistor values will be determined by the filter design equations. By inspection of the "Separate Impedance Tapering" columns of the tables of FIGS. 24 and 25, it follows that, in order to minimize coefficient sensitivity, the resistor tapering factor should actually be held as small as possible. Typically, the resistor tapering factor should be close to unity, while the capacitor tapering factor ρ should be as large as possible.

In FIG. 14, the amplitude response curves for third-order Butterworth and Chebyshev filters are shown. The specifications associated with the filters correspond approximately to those defined above in Tables 1 and 2 above, although the pole Qs of the Chebyshev filter are slightly higher (conforming to the stopband attenuation of 38 dB compared to the 25 dB stopband attenuation of Table 2). The capacitor scaling factor p_(c) was varied from 1 to 5. The resistor scaling factor was determined by the design equations. P SPICE Monte Carlo runs with zero-mean resistors and capacitors and 5% flat distribution were carried out. The response curves of the circuits with tapered capacitors (i.e., ρ_(c) =3 and ρ_(c) =5) are considerably less sensitive to component values than are the non-tapered circuits.

The effect of impendance tapering the nth-order allpole filter of the type shown in FIG. 7 will now be considered. The transfer function is of the form: ##EQU51## or in terms of the individual complex-conjugate pole pairs, for n even ##EQU52## and for n odd ##EQU53##

The relationship between the coefficients in equation 101 and the frequency and Q parameters of equations 102 and 103 are given in the table shown in FIG. 26 for n=2 to 6. For n greater than 3, the analytical computations become increasingly complicated and unwieldy. Furthermore, the nonlinear design equations for a given set of filter specifications and the resulting transfer functions become virtually impossible to solve in most cases for n greater than 3. Nevertheless, assuming the design equations can be solved to determine the R, C and β component values for the circuit, some general observations can be made, which coincide closely with the observations made above regarding filters of second and third order. One observation that can be made is that the general expression set forth in equation 98, ##EQU54## continues to hold, as does the expression relating the variation of amplitude response to changes in coefficient values set forth in equation 34: ##EQU55##

The frequency-dependent functions f_(i) (ω), discussed above, depend only on the coefficients of the initial transfer function. Examples of these functions, which represent the sensitivity of the transfer function (i.e., of the amplitude and, if desired, phase response) to coefficient variations, were given for n=3 and 6 and plotted in FIGS. 4A-5B. It should be noted that in each case the maxima of the curves lie on either side of and close to the passband edge. It is difficult to deduce from these curves whether the amplitude sensitivity to changes of any one particular coefficient is larger than that of the others, although for the cases depicted in FIGS. 4A-5B, the sensitivity functions f_(i) (ω) were seen to increase in the passband and with decreasing i. In any case, the higher the pole Qs of the filter, the larger the individual f_(i) (ω) functions will be (i.e., the higher the amplitude sensitivity to coefficient changes, and hence to component changes will be). As has already been emphasized above, pole Qs increase both with increased ripple and with increased filter order. Therefore, the rule of thumb for the design of low amplitude sensitivity filters is to pick a Butterworth filter with the lowest possible order. In practice, however, the trade-off between filter order and passband ripple may nonetheless come out in favor of a Chebyshev filter. This must be investigated individually for each application and filter specification. However, as stated above, the present invention is not limited to these types of filters, as will be apparent to those skilled in the art after reading the present application.

It has been shown above that impedance tapering filters in accordance with the method of the present invention significantly decreases the frequency response sensitivity to component variations. Unfortunately, the tapering factors cannot be arbitrarily chosen because the degrees of freedom inherent in the design equations may not be sufficient to satisfy both the filter specifications and the selected tapering factors. Clearly, tapering factors that are as large as possible are preferable. In practice, however, the tapering factors may be limited by practical design considerations to values lower than about five. Nevertheless, even this much tapering will decrease the filter sensitivity considerably. It should be noted that the tapering factors are not limited by the present invention but are limited only by practical considerations of filter design, with the exception that the tapering factors must be greater than unity.

Having demonstrated that the sensitivity of filters can be minimized by using the impedance tapering method of the present invention, the design equations for tapered second and third-order filters will now be described. Although the equations for higher order-filters defy a satisfactory closed-form solution, in accordance with the present invention, impedance-tapered second and/or third-order filter sections can be cascaded to provide filters of any desired order which have low sensitivity to component values and gain variations.

DESIGN EQUATIONS FOR THE SECOND ORDER FILTER

The design equations of the present invention used for designing a second-order lowpass filter such as that shown in FIG. 8A, for which the transfer function was given in equations 36 and 37 will now be provided. For convenience, the transfer function is repeated here: ##EQU56## where, instead of the amplifier gain β, the more general factor K is used, and ##EQU57## Referring to FIG. 9 and letting

    R.sub.1 =R C.sub.1 =C

    R.sub.2 =rR C.sub.2 =C/ρ                               (Eq. 109)

and ##EQU58## This results in ##EQU59##

In practice K, a₀ and a₁ or, equivalently, K, ω_(p) and q_(p), will be given by the filter specifications. From these quantities, and possibly some additional constraints, such as input resistance level, maximum or minimum acceptable capacitor values, etc., ω₀, ρ, r and β are determined, as discussed in more detail below.

From equations 111-113, the following expressions are obtained: ##EQU60## r and ρ must be positive, and β, which is the gain of a noninverting opamp, must obey the constraint

    β≧1                                            (Eq. 116)

Thus, the denominator of equations 114 and 115 must be larger than zero, resulting in the constraint, that ##EQU61## Because of equation 116, the expression under the square root in equation 117 will always be positive.

METHOD OF THE PRESENT INVENTION FOR DESIGNING SECOND ORDER FILTERS

Using equations 107-117 above, the second-order filter is designed as follows. First, the transfer function coefficients (a₀, a₁) and the gain (β) for the filter to be designed are obtained from the filter specifications (e.g., from a filter handbook or CAD program). The value of the filter design radian frequency, ω₀, is then obtained from equation 111 using the value of a₀ obtained from the filter specifications. Using equation 114, the value of the resistor tapering factor, r, is calculated using the known values of a₀, ω₀ and β. Next, the value of the capacitor tapering factor, ρ, is calculated using equation 111 and the known values of ω₀, r and a₀. Alternatively, the capacitor tapering factor, ρ, can be calculated using equation 115 and the known values of a₀, ω and β and the value of resistive tapering factor, r, can be calculated using equation 111.

Once these values have been calculated, either R₁ or C₁ is selected in accordance with practical design considerations, which are understood by those of ordinary skill in the art. If the value of R₁ is selected, the value of C₁ is calculated using equation 110. The values for R₂ and C₂ are then obtained using equation 109. Alternatively, the value for C₁ can be selected and then the values of R₁, R₂ and C₂ are calculated using equations 110 and 109, respectively.

DESIGN EQUATIONS FOR THE THIRD ORDER FILTER

The mathematical design equations of the present invention for designing a third-order filter with low sensitivity to component values will now be described with reference to the third-order lowpass filter shown in FIG. 12. The transfer function, as given by equation 70 is repeated here for convenience: ##EQU62## where the more general notation for the gain, K, is used rather than the amplifier gain β, and ##EQU63## Referring to FIG. 13: ##EQU64##

In what follows it will be shown that ω₀ is an important design parameter, the value of which determines the realizability of a third-order filter.

With equations 119 and 121, the following relations between the transfer-function coefficients, tapering factors, design frequency ω₀, and gain are obtained: ##EQU65##

Before discussing the ideal impedance tapering case, i.e., the case for which r₂ =ρ, r₃ =ρ², ρ₂ =ρ, and ρ₃ =ρ², the general case in which R, C, ρ₁, ρ₂, a₀, a₁, a₂ and K are given, and r₂, r₃ and β are found will be discussed. It is useful (but not required) to normalize the coefficients a_(i) with respect to ω₀ =(RC)⁻¹, thus: ##EQU66## and in general ##EQU67## and n is the order of the filter transfer function T(s). After some calculation, the following three equations for the unknown quantities are obtained:

    ar.sub.2.sup.2 +br.sub.2 +c=0                              (Eq. 129)

where

    a=α.sub.0 +α.sub.2 -α.sub.1 -1           (Eq. 130)

    b=α.sub.2 -2                                         (Eq. 131)

    c=-(1+ρ.sub.2)                                         (Eq. 132)

for r₃ : ##EQU68## and for β: ##EQU69## Since r₂ must be positive and real it follows that the discriminant D of the quadratic equation (equation 129) must be greater than zero, thus:

    D=b.sup.2 -4ac =(α.sub.2 -2).sup.2 +4(α.sub.0 +α.sub.2 -α.sub.1 -1)(1+ρ.sub.2)>0                       (Eq. 135)

The necessary and sufficient condition for r₂ to be real and positive results from equation 135 in the form of an upper bound on ω₀. In terms of the transfer function coefficients a_(i), i=0, 1, 2, the following expression is obtained: ##EQU70## Moreover, a realizability condition on the product (r₂ r₃) is shown to also depend on ω₀ max, namely: ##EQU71## This condition implies that for ideal impedance tapering, in which case r₂ r₃ =ρ₂ ρ₃, the following condition must be satisfied: ##EQU72## Whether this condition can be fulfilled depends entirely on the coefficients a_(i) of the specified filter.

Finally, bounds guaranteeing that r₂ be both real and positive, and that the gain β be larger than or equal to unity, are defined by the expression: ##EQU73## where it is understood that the ω₀ contained in α₀ and α₂ obeys the inequality in equation 136.

METHOD OF THE PRESENT INVENTION FOR DESIGNING THIRD ORDER FILTERS

Several examples of the application of the method of the present invention for designing a third order filter in accordance with the design equations set forth above will now be described. Consider the specified amplitude response of a third-order Chebyshev lowpass filter shown in FIG. 15. This is the same as the amplitude response of the Chebyshev lowpass filter shown in FIG. 14. The coefficients of the third-order transfer function (see eq. 118) satisfying these specifications are:

    a.sub.0 =0.749×10.sup.17

    a.sub.1 =0.341×10.sup.12

    a.sub.2 =0.59×10.sup.6                               (Eq. 141)

The corresponding dc gain and pole parameters are:

    K=2

    ω.sub.p =0.5037×10.sup.6

    q.sub.p =1.7

    α=0.295×10.sub.6                               (Eq. 142)

and the maximum design frequency ω₀ max is: ##EQU74## Note that ##EQU75## Thus, from equation 139 it is clear that for the filter given by the coefficients in equation 141, ideal impedance tapering is not possible. This is because ideal impedance tapering would require that r₂ r₃ =ρ₂ ρ₃ in which case, according to equation 139, (ω₀ max)³ /a₀ would have to be larger than unity, which it clearly is not.

The step-by-step design procedure of the present invention for designing the third-order Chebyshev lowpass filter satisfying the specifications given above will now be provided while, at the same time, obeying different types of impedance tapering criteria.

A. IMPEDANCE TAPERING ONLY WITH RESPECT TO THE CAPACITORS

For a capacitively-impedance tapered filter, the step-by-step design is as follows:

1. Calculate ω₀ max

From equations 137 and 141, the following is obtained: ##EQU76## 2. Select ρ₂, ρ₃, ω₀

The selection of these values is influenced by the upper bound discussed above: ##EQU77## Letting ρ₂ =3 and ρ₃ =9 it follows that: ##EQU78## For practical design reasons, which will be understood by persons of ordinary skill in the art, ω₀ is chosen as ω₀ =1.916×10⁵.

3. Calculate α₀, α₁ and α₂ and a, b and c of equation 129.

With ω₀ =1.916×10⁵ the following is obtained: ##EQU79## and therefore:

    a=α.sub.0 +α.sub.2 -α.sub.1 -1=3.439

    b=α-2=1.079

    c=-1-p.sub.2 =-4

4. Calculate r₂ and r₃ :

By solving the quadratic of equation 129 for r₂, the value of r₂ is obtained as:

    r.sub.2 =0.933

From equation 133, ##EQU80## the value of r₃ is determined as

    r.sub.3 =2.718

5. Select C₁ and compute R₁, R₂ and R₃

The value of C₁ is selected as C₁ =900 pF, thus ##EQU81## and:

    R.sub.2 =r.sub.2 R.sub.1 =5.41 kΩ

6. Compute β=K:

From equation 134, the value of β is obtained: ##EQU82##

The resulting circuit is shown in FIG. 16. A simple first-order check for the correctness of these values can be obtained using equation 119 to verify that: ##EQU83## and that the obtained gain:

    β=K

This is readily verified for the circuit of FIG. 16.

The values for the Butterworth filters whose responses are shown in FIG. 14 are given in the table of FIG. 27. The transfer-function coefficients for this filter are:

    a.sub.0 =3.64×10.sup.17

    a.sub.1 =1.02×10.sup.2

    a.sub.2 =1.43×10.sup.6                               (Eq. 146)

The upper limit on ω₀ is: ##EQU84## The design values are obtained in the same step-by-step procedure discussed above for the Chebyshev filters. Therefore, the design procedure will not be repeated here for the Butterworth filter. The effectiveness of impedance tapering the capacitors of the Chebyshev and Butterworth filters in order to reduce the sensitivity to component values is demonstrated in FIG. 14.

The preceding examples demonstrate impedance tapering of the capacitors by starting out with values of ρ_(c) (i.e., ρ₂ and ρ₃ in FIG. 13) and then calculating the resistor values by computing r₂ and r₃. These depend on the choice of the design frequency ω₀ =(R₁ C₁)⁻¹ which must always be smaller than ω₀ max. The latter depends on the desired filter transfer function and its coefficients. As was pointed out earlier, impedance tapering of both the capacitors and the resistors for filters of higher than second order is possible only in rare cases, since the degrees of freedom necessary to satisfy a given set of filter specifications usually permit tapering of only the capacitors or the resistors. As shown in the preceding examples, however, impedance tapering of only the capacitors provides a significant reduction in sensitivity to component values. It shall now be shown that the alternative procedure, i.e., tapering of only the resistors, is effective in the same way. To show this, the Butterworth filter discussed above, whose transfer-function coefficients are given by equation 146 and whose response for various capacitive-impedance tapering values were given in FIG. 14, will be used for purposes of illustration. The amplitude-response specifications are shown in FIG. 17. For this example, a value of K=4 is selected.

B. IMPEDANCE TAPERING ONLY WITH RESPECT TO THE RESISTORS

1. Calculate ω₀ max

Since the filter is being designed in accordance with the coefficients stated in equation 146, ω₀ max is given by equation 147 as ω₀ max =7.1544×10⁵.

2. Select r₂, r₃ and ω₀

The selection of these values is influenced by the upper bound: ##EQU85## Letting r₂ =3 and r₃ =9, the product of ρ₂ and ρ₃ is calculated as: ##EQU86## In accordance with practical design considerations, (in terms of component values) ω₀ is selected to be

    ω.sub.0 =(R.sub.1 C.sub.1).sup.-1 =4.695×10.sup.5(Eq. 150)

3. Calculate α₀, α₁ and α₂ and a, b and c of equation 129

using equation 150, the following values are obtained:

    α.sub.0 =3.5176

    α.sub.1 =4.6276

    α.sub.2 =3.046                                       (Eq. 151)

and therefore

    a=α.sub.0 +α.sub.2 -α.sub.1 -1=0.936

    b=α.sub.2 -2=1.046

    c=-1-ρ.sub.2                                           (Eq. 152)

4. Calculate ρ₂ and ρ₃ :

Solving equations 129 and 152 for ρ₂ the following equation is obtained:

    ρ.sub.2 =ar.sub.2.sup.2 +br.sub.2 -1                   (Eq. 153)

which is a first-order equation which must be solved for capacitance tapering. With the values in equation 152, ρ₂ is obtained as:

    ρ.sub.2 =10.562                                        (Eq. 154)

and with equation 133: ##EQU87## Note that the condition set forth in equation 149 is automatically satisfied if ω₀ <ω₀ max.

5. Select R₁ in accordance with practical design considerations and compute C₁, C₂ and C₃

Select R₁ =10 kΩ, thus ##EQU88## and with equations 154 and 155: ##EQU89## 6. Compute β=K: ##EQU90##

Following this procedure with three representative resistive tapering factors, the three Butterworth filters defined by the values listed in the table of FIG. 28 were obtained. Similarly, for the filter characteristics depicted in FIG. 17 the Chebyshev filter coefficients were obtained:

    a.sub.0 =3.30×10.sup.17

    a.sub.1 =0.91×10.sup.12

    a.sub.2 =1.2×10.sup.6

    K=4                                                        (Eq. 160)

The corresponding filter parameters are also listed in the table of FIG. 28. It can be seen from FIG. 28 that r₂ =r and r₃ =r². The corresponding Monte Carlo runs for 5% component values are shown in FIG. 19. Comparing the reduction in sensitivity to component values for capacitive and resistive impedance tapering, i.e., comparing FIG. 14 with 19, it is evident that both are useful for the design of tolerance-insensitive filter circuits, although tapering only the capacitors (FIG. 14) may be slightly more effective than tapering only the resistors (FIG. 19). The Chebyshev curves in FIG. 19 deteriorate somewhat for r=5 compared to those for r=3. The reason for this is further discussed below.

It has been shown above that in order to find a realizable third-order filter circuit capable of satisfying given filter requirements, the condition ω₀ =(R₁ C₁)⁻¹ <ω₀ max must be satisfied where ω₀ max is given in terms of the filter coefficients as in equation 137. The influence the choice of ω₀ has on the filter design, assuming of course that it is chosen less than ω₀ max, will now be investigated.

To determine the amount of influence the choice of ω₀ has on the component sensitivity of a given circuit, the following procedure of the present invention must be followed. Using the coefficients of equation 141 (which belong to the third-order transfer function satisfying the filter specifications in FIG. 15) as an illustrative example, the procedure can be summarized by the following steps:

1. Select ρ₂, ρ₃ and ω₀ <ω₀ max

Example: for the coefficients of equation 141, ω₀ max =2.9491×10⁵ rad/sec.

Select ρ₂ =3, ρ₃ =9, ω₀ =1.25×10⁵

2. Calculate α₀, α₁, α₂, and a, b, and c

    α.sub.0 =a.sub.0 /ω.sup.3 =38.3488; α.sub.1 =21.824; α.sub.2 =4.72

    a=α.sub.0 +α.sub.2 -a.sub.1 -1=20.2448

    b=α.sub.2 -2=2.72

    c=-ρ.sub.2 -1=-4

3. Calculate r₂ and r₃

From equations 129-132, the following is obtained:

    20.2448r.sup.2.sub.2 +2.72r.sub.2 -4=0

which gives the positive real root

    r.sub.2 =0.3824

and with equation 133:

    r.sub.2 =ρ.sub.2 ρ.sub.3 /r.sub.2 α.sub.0 =1.84

4. Select C₁ and calculate R₁, R₂, R₃, C₂, C₃

Selecting C₁ =900 pF

Thus

    R.sub.1 =1/ω.sub.0 C.sub.1 =8.88 kΩ

and

    R.sub.2 =r.sub.2 R.sub.1 =3.4 kΩ

    R.sub.3 =r.sub.3 R.sub.1 =16.36 kΩ

    C.sub.2 =C.sub.1 /ρ.sub.2 =300 pF

    C.sub.3 =C.sub.1 /ρ.sub.3 =100 pF

These results can be verified to a first order (see equation 145) since ##EQU91## 5. Calculate ⊖=K as follows: ##EQU92## This results in the circuit shown in FIG. 20. Going through the five design steps above for 10 different values of ω₀ (all of which must, of course, be less than ω₀ max), 10 different third-order circuits were determined by the design values listed in the table of FIG. 29. The resulting functions of r₂, r₃. r₃ /r₂ and β vs. the ten values of ω₀ are plotted in FIG. 21. Monte Carlo runs of the 10 resulting third-order lowpass filters for 5% resistors and capacitors are shown in FIG. 22. It should be noted that the tolerances of the filter responses in FIG. 22 become smallest for the value of ω₀ in the region of 26×10⁵. A glance at FIG. 21 shows that this corresponds to a ratio of r₃ /r₂ ≈1, i.e., for the case of r₂ =r₃. This is similar to the conclusion arrived at with second-order networks. Thus, it is preferable to select a value of ω₀ for which r₂ =r₃.

C. IMPEDANCE TAPERING THE CAPACITORS WITH r₂ ≈r₃ =r

Impedance Tapering of the Capacitors with r₂ ≈r₃ =r will now be discussed. As shown above, the design equations 129-134 can be used to design a low-sensitivity filter by selecting the design parameters ω₀, ρ₂ and ρ₃ and then computing the remaining design parameters r₂, r₃ and β. The value for coo can be selected in the manner discussed above, or, alternatively, it can be obtained using the appropriate graph in FIG. 23 for the case where r₂ is approximately equal to r₃. This is done explicitly with the six-step design procedure outlined above. If now r₂ =r₃ is selected, and tapering values for ρ₂ and ρ₃ are selected, equations 129-134 can be used to compute ω₀, r and β. However, because the resulting polynomial equations in ω₀ or r are of sixth-order, and therefore are not directly solvable, these equations cannot besolved in the straight forward manner discussed above. Instead, the bounds on ω₀ can be computed and then the equations can be iteratively solved to obtain, more or less, r₂ ≈r₃ =r. "More or less" is quite sufficient here, since there is no well defined optimum for which the component sensitivities are minimum. The resulting design procedure is summarized in the following design steps. As in one of the previous illustrative examples, the specifications for the third-order Chebyshev filter shown in FIG. 15 shall be used. The corresponding filter coefficients are given by equation 141.

D. ITERATIVE COMPUTATION OF r, r₂ AND r₃

1. Calculate ω₀ max

From equations 137 and 141, the following is obtained:

    ω.sub.0 max =4.sub.a0 /4.sub.a1 -a.sub.2.sup.2 =2.9491×10.sup.5 rad/sec

2. Select ρ₂, ρ₃, and calculate ω₀ max

As in the previous examples, ρ₂ =3 and ρ₃ =9 are selected.

With r₂ =r₃ =r, the following is obtained using equation 123: ##EQU93##

In order to guarantee a minimum of resistive tapering, r>1 is required. The lower bound on ω₀, i.e., ω₀ min, is then obtained for r=1: ##EQU94## Thus, from equation 141: ##EQU95## 3. Select ω₀ between ω₀ min and ω₀ max and calculate r₂ and r₃.

4. Repeat step 3 with new ω₀ until r₂ ≈r₃ =r is found.

The value of ω₀ =2.6×10₅ rad/sec was found using these two steps, for which

    r.sub.2 =2.594

and

    r.sub.3 =2.44

The results of this design are listed in the table of FIG. 29 under (8). It was shown in FIG. 22 that this design yielded the best results, i.e., the lowest sensitivity to component values.

5. Select C₁ and calculate R₁, R₂, R₃, C₂, C₃.

Selecting C₁ =900 pF the following is obtained:

    R.sub.1 =1/ωC.sub.1 =4.27 kΩ

    R.sub.2 =r.sub.2 R.sub.1 =11.07 kΩ

    R.sub.3 =r.sub.3 R.sub.1 =10.42 kΩ

    C.sub.2 =C.sub.1 /ρ.sub.2 =300 ρF

    C.sub.3 =C.sub.1 /ρ.sub.3 =100 ρF                  (Eq. 166)

6.: Calculate β: ##EQU96##

Note that for the component calculations, the actual r₂ and r₃ values obtained in step 4 must be used. The fact that they are close to each other is important and determines the value ω₀ used in the final circuit design.

As stated above, impedance tapered second and/or third order filters which have been impedance tapered in accordance with the method of the present invention can be cascaded to obtain impedance tapered higher order filters. It will be apparent to those skilled in the art how this can be accomplished since cascading lower order filters to obtain higher order filters is generally known in the art of filter design. Therefore, in the interest of brevity, a discussion of this procedure will not be provided.

Reference is now made to FIG. 30, which illustrates a fourth order filter, consistent with the nth order filter design illustrated in FIG. 7. More specifically, FIG. 30 shows a cascaded filter with low sensitivity to component values and gain. As shown, the filter includes a first resistor-capacitor section comprising a first resistor R1 having a first terminal and a second terminal and a first capacitor C1 having a first terminal and a second terminal. The first terminal of the first capacitor C1 is electrically coupled to said second terminal of said first resistor R1. The filter also includes a second resistor-capacitor section electrically coupled to said first resistor capacitor section, wherein the second resistor-capacitor section includes a second resistor R2 having a first terminal and a second terminal and a second capacitor C2 having a first terminal and a second terminal. The first terminal of the second capacitor C2 is electrically coupled to the second terminal of the second resistor R2. The filter also includes a third resistor-capacitor section electrically coupled to said second resistor-capacitor section, wherein the third resistor-capacitor section comprising a third resistor R3 having a first terminal and a second terminal and a third capacitor C3 having a first terminal and a second terminal. The first terminal of the third capacitor C3 is electrically coupled to the second terminal of the third resistor R3. Finally, the filter includes a fourth resistor-capacitor section electrically coupled to said third resistor-capacitor section. The fourth resistor-capacitor section includes a fourth resistor R4 having a first terminal and a second terminal and a fourth capacitor C4 having a first terminal and a second terminal. The first terminal of the fourth capacitor C4 is electrically coupled to said first terminal of the fourth resistor R4, wherein the value C₂ =C₁ ÷ρ, wherein ρ is greater than 1, and wherein C₄ =C₃ ÷ρ₂, wherein ρ₂ is greater than 1. In the illustrated embodiment, R₂ =(R₁ ×r), wherein r is greater than 1 and wherein R₄ =(R₃ ×r₂), wherein r₂ is greater than 1. Further, R₃ =(R₂ ×r₁), wherein r₁ is greater than 1. In addition, C₃ =(C₂ ÷ρ₁), wherein ρ₁ is greater than 1, and C₄ =C₃ ÷ρ₂, wherein ρ₂ is greater than 1.

It should be noted that the present invention has been described with respect to the preferred embodiments but that the present invention is in no way limited to these embodiments. It will also be apparent to those skilled in the art that the concepts and methods of the present invention apply generally to the art of filter design and are not limited to the embodiments discussed. It will also be apparent to those skilled in the art that modifications to the methods discussed above can be made which are within the spirit and scope of the present invention. 

What is claimed is:
 1. A cascaded filter with low sensitivity to component values and gain, said filter comprising:a first resistor-capacitor section comprising a first resistor having a first terminal and a second terminal and a first capacitor having a first terminal and a second terminal, said first resistor having a value R₁, said first capacitor having a value C₁, wherein the first terminal of said first capacitor is electrically coupled to said second terminal of said first resistor; a second resistor-capacitor section electrically coupled to said first resistor capacitor section, said second resistor-capacitor section comprising a second resistor having a first terminal and a second terminal and a second capacitor having a first terminal and a second terminal, said second resistor having a value R₂, said second capacitor having a value C₂, wherein the first terminal of said second capacitor is electrically coupled to said second terminal of said second resistor; a third resistor-capacitor section electrically coupled to said second resistor-capacitor section, said third resistor-capacitor section comprising a third resistor having a first terminal and a second terminal and a third capacitor having a first terminal and a second terminal, said third resistor having a value R₃, where R₃ =(R₂ ×r₁), where r₁ is greater than 1, said third capacitor having a value C₃ wherein the first terminal of said third capacitor is electrically coupled to the second terminal of said third resistor; and a fourth resistor-capacitor section electrically coupled to said third resistor-capacitor section, said fourth resistor-capacitor section comprising a fourth resistor having a first terminal and a second terminal and a fourth capacitor comprising a first terminal and a second terminal, said fourth resistor having a value R₄, said fourth capacitor having a value C₄, wherein the first terminal of said fourth capacitor is electrically coupled to said first terminal of said fourth resistor, wherein the value C₂ =C₁ ρ, wherein ρ is greater than 1, and wherein C₄ =C₃ /ρ₂, wherein ρ₂ is greater than
 1. 2. The filter of claim 1 wherein R₂ =(R₁ ×r), wherein r is greater than 1 and wherein R₄ =(R₃ ×r₂) wherein r₂ is greater than
 1. 3. The filter of claim 1 wherein C₃ =(C₂ ÷ρ₁), wherein ρ₁ is greater than
 1. 4. A cascaded filter with low sensitivity to component values and gain, said filter comprising:a first resistor-capacitor section comprising a first resistor having a first terminal and a second terminal and a first capacitor having a first terminal and a second terminal, said first resistor having a value R₁, said first capacitor having a value C₁, wherein the first terminal of said first capacitor is electrically coupled to said second terminal of said first resistor; a second resistor-capacitor section electrically coupled to said first resistor capacitor section, said second resistor-capacitor section comprising a second resistor having a first terminal and a second terminal and a second capacitor having a first terminal and a second terminal, said second resistor having a value R₂, said second capacitor having a value C₂, wherein the first terminal of said second capacitor is electrically coupled to said second terminal of said second resistor; a third resistor-capacitor section electrically coupled to said second resistor-capacitor section, said third resistor-capacitor section comprising a third resistor having a first terminal and a second terminal and a third capacitor having a first terminal and a second terminal, said third resistor having a value R₃, where R₃ =R₂ ×r₁ and where r₁ is greater than 1, said third capacitor having a value C₃, wherein the first terminal of said third capacitor is electrically coupled to the second terminal of said third resistor; and a fourth resistor-capacitor section electrically coupled to said third resistor-capacitor section, said fourth resistor-capacitor section comprising a fourth resistor having a first terminal and a second terminal and a fourth capacitor comprising a first terminal and a second terminal, said fourth resistor having a value R₄, said fourth capacitor having a value C₄, wherein the first terminal of said fourth capacitor is electrically coupled to said first terminal of said fourth resistor, wherein R₂ =R₁ ×r, wherein r is greater than 1, and wherein R₄ =R₃ ×r₂ wherein r₂ is greater than
 1. 5. The filter of claim 4 wherein C₂ =C₁ ÷ρ, wherein ρ0 is greater thin
 1. 6. The filter of claim 5 wherein C₃ =C₂ ÷ρ₁, wherein ρ₁ is greater than
 1. 7. The filter of claim 7 wherein C₄ =C₃ ÷ρ₂, wherein ρ₂ is greater than
 1. 8. A method of designing a cascaded filter with low sensitivity to component values comprising the steps of:designing a first filter circuit having a first filter section and a second filter section, said first filter section comprising a first resistor having a value R and a first capacitor having a value C, said second filter section comprising a second resistor having a value R₂ and a second capacitor having a value C₂ wherein said second filter section has a higher impedance than said first filter section; designing a second filter circuit having a first filter section and a second filter section, said first filter section of said second filter circuit comprising a third resistor having a value R₃, where R₃ =R₂ ×r₁, and where r₁ is greater than 1, and a third capacitor having a value C₃, said second filter section of said second filter circuit comprising a fourth resistor having a value R₄, and a fourth capacitor having a value C₄, wherein said second filter section of said second filter circuit has a higher impedance than said first filter section of said second filter circuit; and cascading said first and second filter circuits.
 9. The method of claim 8 wherein r₁ is less than or equal to
 10. 10. The method of claim 8 wherein R₂ =R×r, wherein r is greater than
 1. 11. The method of claim 10 wherein r is less than or equal to
 10. 12. The method of claim 8 wherein R₄ =R₃ ×r₂, wherein r₂ is greater than
 1. 13. The method of claim 12 wherein r₂ is less than or equal to
 10. 14. The method of claim 8 wherein C₂ =C÷ρ, wherein ρ is greater than
 1. 15. The method of claim 14 wherein ρ is less than or equal to
 10. 16. The method of claim 14 wherein C₃ =C₂ ÷ρ₁, wherein ρ₁ is greater than
 1. 17. The method of claim 16 wherein ρ₁ is greater than or equal to
 10. 18. The method of claim 16 wherein C₄ =C₃ ÷ρ₂, wherein ρ₂ is greater than
 1. 19. The method of claim 18 wherein ρ₂ is greater than or equal to
 10. 